A rolling ball has total kinetic energy \(100 \mathrm{J}, 40 \mathrm{J}\) of which is rotational energy. Is the ball solid or hollow?

Kinetic energy refers to the energy that an object possesses due to its motion. It can be calculated using the formula \( KE = \frac{1}{2} mv^2 \) where \( m \) represents the mass of the object and \( v \) is its velocity. The concept of kinetic energy is fundamental in physics since it translates to the work that is done to accelerate an object and the work the object can do when it slows down.

In the context of our rolling ball example, the total kinetic energy combines both rotational and translational forms, which is what allows the ball to move and roll. The overall energy is partitioned between these forms based on the ball's characteristics, such as its mass, radius, and whether it is solid or hollow.

Translational kinetic energy is the portion of kinetic energy associated with an object's linear motion. When a ball rolls on a surface without slipping, its center of mass moves linearly in space. In this scenario, translational kinetic energy is given by the same formula for kinetic energy based solely on the linear motion, \( KE_{trans} = \frac{1}{2} mv^2 \).

However, in the case of the rolling ball discussed in the exercise, its motion includes both translation and rotation. Energy conservation tells us that the total kinetic energy is the sum of its translational and rotational parts, which is a crucial concept when analyzing the dynamics of rolling objects.

The moment of inertia is a property of a body that measures its resistance to angular acceleration, similar to how mass measures resistance to linear acceleration. It is dependent on the distribution of an object's mass relative to the axis about which it rotates. Mathematically, it's represented by the equation \( I = \sum mr^2 \), where \( m \) is the mass of a point on the object and \( r \) is the distance from that point to the axis of rotation.

Different Shapes, Different Moments

For objects like spheres, cylinders, or discs, the moment of inertia has standard formulas depending on shape. As seen in the exercise, the value of the moment of inertia plays a role in determining if a ball is solid or hollow, which is essential information when trying to compute an object's rotational kinetic energy.

Understanding the physics of rolling objects involves dissecting how rotation and translation interact. A purely rolling motion implies that there is no slipping between the rolling object and the surface. This condition leads to a relationship where the linear velocity at the object's surface relative to the center of mass is zero.

When accounting for the kinetic energies, the total kinetic energy (TE) of a rolling object is the sum of its translational kinetic energy (\(KE_{trans}\)) and rotational kinetic energy (\(KE_{rot}\)), expressed as \(TE = KE_{trans} + KE_{rot}\).

By using conservation laws and understanding the ratio between rotational and translational kinetic energies, it is possible to determine characteristics of the rolling object, such as whether it is solid or hollow, based on given energy values—and this is precisely what the exercise illustrates by contrasting the energy fractions to ascertain the ball's structure.